Uniformly and strongly consistent estimation for the Hurst function of a Linear Multifractional Stable Motion

Abstract

Since the middle of the 90s, multifractional processes have been introduced for overcoming some limitations of the classical Fractional Brownian Motion model. In their context, the Hurst parameter becomes a Hölder continuous function $H(\cdot)$ of the time variable $t$. Linear Multifractional Stable Motion (LMSM) is the most known one of them with heavy-tailed distributions. Generally speaking, global and local sample path roughness of a multifractional process are determined by values of its parameter $H(\cdot)$; therefore, since about two decades, several authors have been interested in their statistical estimation, starting from discrete variations of the process. Because of complex dependence structures of variations, in order to show consistency of estimators one has to face challenging problems.

The main goal of our article is to introduce, in the setting of the symmetric $\alpha$-stable non-anticipative moving average LMSM, where $\alpha\in(1,2)$, a new strategy for dealing with such kind of problems. It can also be useful in other contexts. In contrast with previously developed strategies, this new one does not require to look for sharp estimates of covariances related to functionals of variations. Roughly speaking, it consists of expressing variations in such a way that they become independent random variables up to negligible remainders. Thanks to it, we obtain, an almost surely and $L^{p}(\Omega)$, $p\in(0,4]$, consistent estimator of the whole function $H(\cdot)$, which converges, uniformly in $t$, and even for some Hölder norms. Also, we obtain estimates for the rates of convergence. Such kind of strong consistency results in uniform and Hölder norms are rather unusual in the literature on statistical estimation of functions.